Let f:G→G be a strictly piecewise monotone continuous map on a finite graph G. By investigating the topological structure of the inverse limit space (G,f) using f as a sole bonding map, we show that the following statements are equivalent: (1) (G,f) contains no indecomposable subcontinuum. (2) The topological entropy of f is zero. (3) (G,f) is Suslinean. (4) Each homeomorphism of (G,f) has zero topological entropy. (5) f has finitely many nontrivial minimal sets. (6) The set of recurrent points of f is closed. (7) Each ω-limit point is a recurrent one. (8) Each recurrent point of f is an almost periodic one.